Exact Solution for Three-Dimensional Ising Model

Three-dimensional Ising model in zero external field is exactly solved by operator algebras, similar to the Onsager's approach in two dimensions.The partition function of the simple cubic crystal imposed by the periodic boundary condition along both (0 1 0) and (0 0 1) directions and the screw boundary condition along the (1 0 0) direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. When the z axis is chosen as the transfer matrix direction, a order-disorder transition in the infinite crystal occurs at a temperature T=Tc determined by the condition: sin 2J/kBTc sinh 2(J1+J2)/kBTc12J) are the interaction energies in three directions, respectively. The analytical expressions for the internal energy and the specific heat are also given.It is also shown that the thermodynamic properties of 3D Ising model with J1=J2 are connected to those in 2D Ising model with the interaction energies (J1 J2D) by the relation (J2D/kBT)*=(J/kBT)*-J1/kBT, ,where x*=1/2lncothx=tanh-1(e-2x).